In some applications (such as time-varying waveguide reverberation
[430]), it may be preferable to compensate for the power
modulation so that changes in the wave impedances of the waveguides do
not affect the power of the signals propagating within.

In [432,433], three methods are discussed for making signal
power invariant with respect to time-varying branch impedances:

The normalized waveguide scheme compensates for power
modulation by scaling the signals leaving the delays so as to give
them the same power coming out as they had going in. It requires two
additional scaling multipliers per waveguide junction.

The transformer-normalized waveguide approach changes the
wave impedance at the output of the delay back to what it was at the
time it entered the delay using a ``transformer'' (defined in
§C.16).

The transformer-normalized DWF junction is shown in Fig.C.27
[432]. As derived in §C.16, the transformer ``turns
ratio'' is given by

We can now modulate a single scattering junction, even in arbitrary
network topologies, by inserting a transformer immediately to the left
or right of the junction. Conceptually, the wave impedance is not
changed over the delay-line portion of the waveguide section; instead,
it is changed to the new time-varying value just before (or after) it
meets the junction. When velocity is the wave variable, the
coefficients and in Fig.C.27 are swapped
(or inverted).

So, as in the normalized waveguide case, for the price of two extra
multiplies per section, we can implement time-varying digital filters
which do not modulate stored signal energy. Moreover, transformers
enable the scattering junctions to be varied independently, without
having to propagate time-varying impedance ratios throughout the
waveguide network.

It can be shown [433] that cascade waveguide chains built using
transformer-normalized waveguides are equivalent to those
using normalized-wave junctions. Thus, the transformer-normalized DWF
in Fig.C.27 and the wave-normalized DWF in Fig.C.22 are
equivalent. One simple proof is to start with a transformer
(§C.16) and a Kelly-Lochbaum junction (§C.8.4),
move the transformer scale factors inside the junction, combine terms,
and arrive at Fig.C.22. One practical benefit of this
equivalence is that the normalized ladder filter (NLF) can be
implemented using only three multiplies and three additions instead of
the usual four multiplies and two additions.